414 XV NORMED ALGEBRAS AND SPECTRAL THEORY linear form x*-+(T • x \y) is continuous on dom(T); when this is so, this linear form extends by continuity to the whole of E (5.5.4) and therefore can be written as xi~-+(x\T* • y) for a uniquely determined vector T* • y, because dom(T) is dense in E (6.3.2). This uniqueness shows that T* is a linear map- ping of F into E, and therefore an unbounded operator, called the adjoint of T. (When dom(T) = E and T is continuous, this definition clearly agrees with that of (11.5).) Hence we have (15.12.3.1) (T-x\y) = (x\T*-y) for all x e dom(T) and y e dom(T*). If Tl is an unbounded operator such that dom^) ID dom(T), then dom(T*) c dom(r*). In what follows we shall endow E x E with the structure of & Hilbert space such that ((*!» *2> I Ol> 72)) = Ol I Jl) + (*2 I J>2)» so that E x E is the Hilbert sum of its two subspaces E x {0} and {0} x E, each of which is isomorphic to E. Also we denote by /the continuous operator (x,y)\-^(yy —x); clearly J is a unitary operator on E x E, and J2 — —I. (15.12.4) Let T be an unbounded operator on E, such that dom( T) is dense in E. (i) The adjoint operator T* is closed, and its graph F( T*) is the orthogonal supplement ofJ(T(T)) in the Hilbert space E x E. (ii) The following properties are equivalent: (a) T can be extended to a closed operator. (b) dom(T*) is dense in E. If these conditions are satisfied, the graph of any closed operator extending T contains the graph of T**, and F(T**) = F(T). (Thus T** is the smallest closed operator which extends T, and in particular, T** = T if T is closed.) Moreover, (T**)* = T*. (i) If a sequence (yn) of points in dom(T*) converges to yeE and is such that the sequence (T* - yn) converges to zeE, then the sequence of continuous linear forms x H~> (x \ T* • yn) converges for all jteE to the continuous linear form xt-*(x\z). But if ^edom(T), we have (x\z) =lim(T'X\yn) = (T-x\y), hence jedom(T*) and z=T*->> by »-*• oo definition, which shows that T* is closed. On the other hand, to say that (y, z) e E x E is orthogonal to all the vectors (T - x, —x) with x e dom(T) signifies that (T - x \ y) = (x \ z), i.e., that x h-» (T • x | y) is continuous, hence y e dom( T*) and z = T* - y.hat the